Cohomology of flag supervarieties and resolutions of determinantal ideals. II

TL;DR

This paper computes the coherent cohomology of complex periplectic Grassmannians, revealing a tensor product structure involving classical cohomology rings and representations of the periplectic Lie supergroup, with implications for determinantal ideals.

Contribution

It introduces methods to compute cohomology of periplectic Grassmannians and links these results to syzygies of determinantal ideals, advancing understanding of supervarieties.

Findings

Cohomology decomposes into tensor products involving classical cohomology and supergroup representations.

Explicit description of the supergroup representation restriction to its even subgroup.

Development of tools for studying splitting rings of Coxeter groups of types BC and D.

Abstract

We compute the coherent cohomology of the structure sheaf of complex periplectic Grassmannians. In particular, we show that it can be decomposed as a tensor product of the singular cohomology ring of a Grassmannian for either the symplectic or orthogonal group together with a semisimple representation of the periplectic Lie supergroup. The restriction of the latter to its even subgroup has an explicit multiplicity-free description in terms of Schur functors and is closely related to syzygies of (skew-)symmetric determinantal ideals. We develop tools for studying splitting rings for Coxeter groups of types BC and D, which may be of independent interest.